Engineering Voltage gain for amplifier (JFET) in common gate

AI Thread Summary
The discussion focuses on calculating the voltage gain (A_V) for a JFET amplifier in a common gate configuration using the hybrid model. The user derives an expression for A_V but notices a discrepancy with the teacher's answer, specifically the absence of the term R_S in the denominator of their equation. The user suspects that their assumption about the input capacitor behaving as a short circuit may be the reason for this difference. They seek clarification on how R_S factors into the overall gain equation. The conversation highlights the importance of accurately accounting for all components in amplifier circuit analysis.
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Homework Statement
Prove that the voltage gain for the following amplifier in common gate is $$A_V=\frac{(g_mr_d+1)R_L}{R_S(g_mr_d+1)+R_L+r_d}$$
Relevant Equations
* We use the Hybrid model to analyze the transistor.

$$A_V=\frac{(g_mr_d+1)R_L}{R_S(g_mr_d+1)+R_L+r_d}$$
Screenshot from 2021-12-18 17-19-09.png


First the hybrid model, I assume the capacitor works as a short circuit regarding the altern current:
Screenshot from 2021-12-18 17-24-05.png


$$A_V=\frac{v_o}{v_i}$$
$$v_i=-v_{GS}$$
$$v_o=-i_LR_L$$
$$i_L=g_mv_{GS}+\frac{v_{DS}}{r_d}$$

Now I use the Kirchhoff's law to get $v_{DS}$. I consider this close loop:
Screenshot from 2021-12-18 17-58-04.png


$$-v_i-v_{DS}+v_o=0$$
$$v_{DS}=-v_i+v_o=v_{GS}+v_o$$
$$i_L=g_mv_{GS}+\frac{v_{GS}+v_o}{r_d}$$
$$v_o=-i_LR_L=-\left(g_mv_{GS}+\frac{v_{GS}+v_o}{r_d}\right)R_L$$
Now I just try to get $v_o$ from that equation:
$$v_o=-g_mv_{GS}R_L-\frac{v_{GS}+v_o}{r_d}R_L$$
$$v_o=-g_mv_{GS}R_L-\frac{v_{GS}}{r_d}R_L-\frac{v_o}{r_d}R_L$$
$$v_o+\frac{v_o}{r_d}R_L=-g_mv_{GS}R_L-\frac{v_{GS}}{r_d}R_L$$
$$v_o\left(1+\frac{R_L}{r_d}\right)=-g_mv_{GS}R_L-\frac{v_{GS}}{r_d}R_L$$
$$v_o=\frac{-g_mv_{GS}R_L-\frac{v_{GS}}{r_d}R_L}{1+\frac{R_L}{r_d}}$$
$$v_o=\frac{-v_{GS}R_L\left(g_m+\frac{1}{r_d}\right)}{1+\frac{R_L}{r_d}}$$
$$v_o=\frac{v_iR_L\left(g_m+\frac{1}{r_d}\right)}{1+\frac{R_L}{r_d}}$$
$$A_V=\frac{v_o}{v_i}=\frac{R_L\left(g_m+\frac{1}{r_d}\right)}{1+\frac{R_L}{r_d}}\cdot\frac{r_d}{r_d}$$
$$\boxed{A_V=\frac{R_L\left(g_mr_d+1\right)}{r_d+R_L}}$$

The answer given by my teacher is:
$$A_V=\frac{(g_mr_d+1)R_L}{R_S(g_mr_d+1)+R_L+r_d}$$

As you can see I am missing a whole term in the denominator. I do not understand how ##R_S## gets into the equation.
 
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